Peace be upon you,
In the question A late-diverging "approximating solution" for a system of functional equations, I have asked for an approximating solution for a system of functional equations.
Here, I have presented some alternative forms of that system, to be analytically solved.
Since the real part of the domains of its psi functions are positive we can write[1] it as \begin{align*} \psi(s)=-\gamma + \int_0^1\frac{1-x^{s+1}}{1-x} \end{align*} where $γ$ is the Euler-Mascheroni constant. So, we can rewrite our system to become \begin{align*} \begin{cases} \int_0^1\frac{x^{\alpha+\beta+1}-x^{\alpha+1}}{1-x}=c_1\\ \int_0^1\frac{x^{\alpha+\beta+1}-x^{\beta+1}}{1-x}=c_2 \end{cases} \end{align*} Also, using the Newton series we can write[2] \begin{align*} \psi(s)=-\gamma-\sum_{k=0}^{\infty}{\frac{(-1)^k}{k}\binom{s+1}{k}} \end{align*} So, we can rewrite our system to become \begin{align*} \begin{cases} \sum_{k=0}^{\infty}{\frac{(-1)^k}{k}\binom{\alpha+\beta+1}{k}}-\sum_{k=0}^{\infty}{\frac{(-1)^k}{k}\binom{\alpha+1}{k}}=c_1\\ \sum_{k=0}^{\infty}{\frac{(-1)^k}{k}\binom{\alpha+\beta+1}{k}}-\sum_{k=0}^{\infty}{\frac{(-1)^k}{k}\binom{\beta+1}{k}}=c_2 \end{cases} \end{align*} Also, using the Recurrence formula and characterization we can write[3] \begin{align*} \psi(s)=-\gamma+\sum_{k=1}^{\infty}\left(\frac{1}{k}-\frac{1}{s+k+1}\right) \end{align*} So, we can rewrite our system to become \begin{align*} \begin{cases} \sum_{k=1}^{\infty}\left(\frac{1}{k}-\frac{1}{\alpha+\beta+k+1}\right)-\sum_{k=1}^{\infty}\left(\frac{1}{k}-\frac{1}{\alpha+k+1}\right)=c_1\\ \sum_{k=1}^{\infty}\left(\frac{1}{k}-\frac{1}{\alpha+\beta+k+1}\right)-\sum_{k=1}^{\infty}\left(\frac{1}{k}-\frac{1}{\beta+k+1}\right)=c_2 \end{cases} \end{align*} Is there any hint for "analytically solving" one of the aforementioned systems using?